where
is the amplitude (in arbitrary units),
is the
phase in radians, and
is the frequency in radians per
second. Time
is a real number that varies
continuously from minus infinity to infinity in the ideal
sinusoid. All three parameters
are real numbers.
In addition to radian frequency
, it is useful to define
, where
is the frequency in
Hertz (Hz).6.1

where
is the delta function or impulse
at frequency
(see Fig.5.4 for a plot, and
§B.10 for a mathematical introduction).
Since the delta function is even (
),
we can also write
. It is shown in §B.13 that the
sinc
limit
above approaches a delta function
.
However, we will only use the Discrete Fourier Transform (DFT)
in any practical applications, and in that case, the result is easy to
show [264].

The inverse Fourier transform is easy to evaluate by the sifting
property6.3of delta functions:

(6.6)

Substituting into (5.4), the spectrum of our original sinusoid
is given by

(6.7)

which is a pair of impulses, one at frequency
having
complex amplitude
, summed with another at
frequency
with complex amplitude
.